On the Number of Lattice Free Polytopes

V. I. Arnold asked for estimates for the number of equivalence classes of lattice polytopes, under the group of unimodular affine transformations. What we investigate here is the analogous question for lattice free polytopes. Some of the results: the number of equivalence classes of lattice free simplices of volume at most v in dimension d is of order vd − 1, and the number of equivalence classes of lattice free polytopes of volume at most v in dimension d isO (v2d − 1(logv )d − 2).